Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Flow reversals of a non-Newtonian fluid in an expanding channel

Abstract: The flow of an incompressible power-law fluid through convergent-divergent channels is considered where the choice of the viscosity is such that the zero-shear rate viscosity is neither zero nor infinity for any finite value of the power-law exponent. Rather than employing the classical similarity transformation employed by Jeffery and Hamel implying purely radial solutions for the velocity field, we instead consider flow in both the radial and angular directions, under three sets of boundary conditions. This work is in contrast to the earlier study by Mansutti and Rajagopal (1991). wherein the viscosity could be zero or infinity for certain values of the power-law exponent. While seeking solutions to nonlinear differential equations, when seeking similarity solutions, one ought to be cognizant that the equations might have solutions in addition to the similarity solution that is being sought. We observe that the tangential velocity does indeed play a role in the flow regime of the fluid, with flow reversal also present for certain values of α. In the case of traction boundary conditions, we also observe a change in the flow characteristics.

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

Abstract: In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.

NOTES AND CORRESPONDENCE Internal Boundary Layer Scaling in ''Two Layer'' Solutions of the

Abstract: The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline equations in the limit of small vertical diffusivity ky , it must contain an internal boundary layer that collapses to a discontinuity as ky → 0. An asymptotic internal boundary layer equation is derived for this case, and the associated boundary layer thickness is proportional to . In general, the boundary layer remains three-dimen1/2 ky sional and the thermodynamic equation does not reduce to a vertical advective‐diffusive balance even as the boundary layer thickness becomes arbitrarily small. If the vertical convergence varies sufficiently slowly with horizontal position, a one-dimensional boundary layer equation does arise, and an explicit example is given for this case. The same one-dimensional equation arose previously in a related analysis of a similarity solution that does not itself approach a two-layer solution in the limit ky → 0. 1. Internal boundary layer scaling Stommel and Webster (1962) discovered a similarity solution of the thermocline equations with an internal boundary layer that could be interpreted as a model of the subtropical main thermocline. The internal boundary layer marks the base of the wind-driven motion, as the deeper circulation is driven by vertical diffusion of heat through the internal boundary layer. The characteristic thickness of the Stommel‐Webster internal boundary layer is , where ky is a constant vertical diffusivity. 1/2 ky Originally obtained by a linearized analysis, this scaling was confirmed by Young and Ierley (1986) using matched asymptotic expansions. It contrasts with the thickness dependence that follows from the tradi1/3 ky tional advective‐diffusive scaling of the thermocline equations (Welander 1971). The boundary layer is evidently not peculiar to 1/2 ky the particular similarity form studied by Stommel and Webster (1962) and Young and Ierley (1986). Salmon (1990) showed that a internal boundary layer should 1/2 ky generally arise in subtropical-gyre solutions of the thermocline equations, although this result depended on a Taylor-series argument that itself relied on an assump

A similarity solution for light oil spreading on groundwater

Abstract: Spreading of a LNAPL lens (oil) in the vicinity of the groundwater table in a 2-D planar or axisymmetric homogeneous domain is described using a multiphase flow model including capillary forces and oil entrapment by water. Assuming vertical flow equilibrium the equations are vertically integrated. Hence, the free oil volume per unit lateral area and the vertically averaged relative permeability are explicitly related to the vertical position of the interface between zones with either two or three phases. The trapped oil volume is approximated by a linear hysteretic relation that is based on the maximum in time of the free oil volume. The resulting nonlinear diffusion equation admits a similarity solution for the free oil volume per unit lateral area as a function of time and the lateral space coordinate. Additionally, expressions for the extension and amount of trapped oil are obtained. The applicability of the analytical solution is demonstrated by comparison to a simulation, which is based on the nonreduced flow model. Special attention is paid to criteria to determine the remaining volume parameter and the time scale of the similarity solution. After determination of these parameters the analytical solution is a good approximation of the spreading at all larger times.

Nonequilibrium vibrational energy transfer across rarefied hypersonic boundary layers

Abstract: The transport of nonequilibrium-excited molecular vibration energy across rarefied hypersonic laminar boundary layers is studied analytically for the case where a finite rate of vibrational energy quenching is allowed on the body surface in the presence of both energy and temperature slip.. This work finds application, for example, in contemporary experimental/computational studies of low density hypersonic vehicle flow fields. Using a local similarity solution approach, explicit closed-form universal relationships are obtained for the nonequilibrium vibrational heating and surface conditions on blunt nosed and slender hypersonic bodies with an arbitrary wall catalysis rate including first order slip effects.